The cylindrical billiard knot, commonly called
"Turk's head knot", is the knot obtained
from a closed trajectory of a ball in a cylindrical billiard (we assume
that the ball is not subject to gravity!) by modifying the self-intersection
points and rather passing alternatively above and below an existing braid.

We talk of a knot of type (q x p) or (p,q)
(p and
q are coprime) if the ball rebounds p times
on each edge and turns q times around the axis (knot experts call
them Turk's head knot with pbights, or braids,
and
q leads, or strands or even coils). Between two
rebounds, the curve self-intersects q – 1 times: therefore, there
are a total of p (q – 1) crossings.Parametrization (portions of circular
helices): .

Here, p = 7 and q =3; 7 rebounds on each horizontal edge,
3-1 = 2 crossings on a trajectory between two edges.

With a conical projection on a plane perpendicular to
the axis, the cylindrical sine wave is flattened into a polygasteroid
with parametrization:; if the initial version is projected, then the portions
of circular helices are projected onto portions of reciprocal
spirals:.

When p > 2q, but only in this case, the
curve is topologically equivalent to the crossed polygon or polygram
with Schläfli symbol {p/q} (p vertices, with every q-th
point connected to one another).

On the left, in the case (3x4), passages above or below
were alternated.But if, while going from one edge to the over, we always
go above, and if, during the following passage between two edges, we always
go below, then we get a toric
knot, that differs from the alternated knot as soon as .

Given p and q two non-coprime integers,
define d = gcd(p, q), p' = p/d, q' = q/d ;
the trajectory of a cylindrical billiard of type (q' x p')
and its d – 1 images by the rotations by
around the axis of the cylinder, with alternated above/below crossings,
form a link with
d components,
also called Turk's head of type (q x p).There still are p (q – 1) crossings.As we will see on the pictures, these links are very
popular in Islamic art.

For q = 1 or 2, the link is equivalent to the toric
link of the same type.

See also the graph
associated to the Turk's head of type (q x p) (which
has p(q–1) edges), that enables an easy drawing of the knot.Opposite, the type (3x5) and (4x5) knots, and their graphs
with 10 and 15 edges.